Ways Modality Could Be: Revised and Expanded

My article "Ways Modality Could Be: Revised and Expanded" is now up on Scholardarity.

Here's an excerpt:

1. Introduction

            In this paper I introduce the idea of a higher-order modal logic—not a modal logic for higher-order predicate logic, but rather a logic of higher-order modalities. “What is a higher-order modality?”, you might be wondering. Well, if a first-order modality is a way that some entity could have been—whether it is a mereological atom, or a mereological complex, or the universe as a whole—a higher-order modality is a way that a first-order modality could have been. First-order modality is modeled in terms of a space of possible worlds—a set of worlds structured by an accessibility relation, i.e., a relation of relative possibility—each world representing a way that the entire universe could have been. A second-order modality would be modeled in terms of a space of spaces of (first-order) possible worlds, each space representing a way that (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way that things actually are, there is a unique actual space which represents the way that first-order modality actually is.

            One might wonder what the accessibility relation itself is like. Presumably, if it is logical or metaphysical modality that is being dealt with, it is reflexive; but is it also symmetric, or transitive? Especially in the case of metaphysical modality, the answer is not clear. And whichever of these properties it may or may not have, could that itself have been different? Could at least some rival modal logics represent different ways that first-order modality could have been?

            To be clear, the idea behind my proposal is not just that some things which are possible or necessary might not have been so at the first order, as determined by the actual accessibility relation, but also that the actual accessibility relation, and hence the nature or structure of actual modality, could have been different at some higher order of modality. Even if the accessibility relation is actually both symmetric and transitive, perhaps it could (second-order) have been otherwise: There is a (second-order) possible space of worlds in which it is different, where it fails to be symmetric, or transitive. We must, therefore, introduce the notion of a higher-order accessibility relation, one that in this case relates spaces of first-order worlds. The question then arises as to whether that relation is symmetric, or transitive. We can then consider third-order modalities, spaces of spaces of spaces of possible worlds, where the second-order accessibility relation differs from how it actually is. I can see no reason why there should be a limit to this hierarchy of higher-order modalities, any more than I can see a reason why there should be a limit to the hierarchy of higher-order properties. There will thus be an infinity of orders, one for each positive integer, and each order will have an accessibility relation of its own. To keep things as clear as possible, a space of first-order points (i.e., of possible worlds) shall be called a galaxy, a space of second-order points, a universe, and a space of any higher order, a cosmos. However, to keep things as simple as possible, in what follows I will deal with but a single cosmos at a time, and hence will not deal with modalities higher than the third order.

            The accessibility relation is not the only thing that might be thought to vary between spaces of worlds: Perhaps the contents of the spaces can vary as well. While I presume that the contents of the worlds themselves remain constant—it makes doubtful sense to suppose that in one space some entity e exists in a world w and in another space e doesn’t exist in that same world w—we may suppose that different spaces may differ as to which worlds they contain, just as different worlds may differ as to which objects they contain. Thus we might have a higher-order analogue of a variable-domain modal logic. There seem, then, to be three ways in which spaces can differ: First, as to the properties of the accessibility relation; second, as to which worlds the relation relates; and third, as to which worlds or spaces are parts of their domains.

            The paper will be structured as follows. In Section 2 I provide some reasons why one might want to pursue this kind of project in the first place. In Section 3 I outline the syntax and semantics of my proposed logic. Section 4 covers semantic tableaux for this system; and after giving the rules for their construction, I construct a few of them myself to establish some logical consequences of the system and give the reader a feel for how it works. In Section 5 I outline a potential application of my framework to the metalogic of modal logics. In Sections 6, 7 and 8 I explore some of  its potential philosophical implications for areas besides logic, namely the philosophy of language; metaphysics, including the metaphysics of modality, the philosophy of time, and laws of nature; and finally the philosophy of religion, before concluding the paper in Section 9.

A Brief Sketch of Kant’s Critical Philosophy

Over at Scholardarity I've posted A Brief Sketch of Kant’s Critical Philosophy in Scholardarity Student’s subsection Open Source Study Notes. Here's an excerpt:

Metaphysical knowledge can be either dogmatic or critical. Dogmatic metaphysics seeks to know things as they are in themselves. Critical metaphysics, which Kant calls “Critique”, only gives us knowledge of things as they must appear to us, and hence of the necessary features of all possible experience. Dogmatic metaphysics would have to meet two requirements which are inconsistent in Kant’s system. First, it would have to be synthetic a priori. It could not be analytic a priori, for then it could not give us new knowledge. Neither could it be synthetic a posteriori, for then it could tell us no more than natural science does. Second, it would have to go beyond the bounds of all possible experience; otherwise, it would not be distinct from mathematics and geometry, which, while also synthetic a priori, are limited to possible experience. This limitation is what makes them possible, for as we said above, they are “built into” space and time as forms of our sensibility. Anything which can appear to us must be subject to our forms of sensibility, and so mathematics and geometry must hold of all appearances. But since dogmatic metaphysics is supposed to apply to things which cannot appear to us, we cannot know a priori what they are like, for they are not subject to the only conditions under which experience, and hence synthetic a priori knowledge, is possible. In consequence, metaphysical knowledge of a dogmatic sort is impossible. Now we can see the source of Kant’s distaste for dogmatic metaphysics: It poses questions which it cannot answer.

Objectivism and The Corruption of Rationality–Formatting Fixed

The formatting of Scott Ryan’s book Objectivism and The Corruption of Rationality, available in the Epistemology section of Scholardarity, has been fixed and now coincides with the print version.

Objectivism and the Corruption of Rationality -- Now available for free at Scholardarity.com

As co-founders of Scholardarity.com, Peter Krey and Jason Zarri are happy to announce that Scott Ryan’s book Objectivism and the Corruption of Rationality: A Critique of Ayn Rand’s Epistemology is now available for free at Scholardarity.com, in the Epistemology section.

Description:

Ayn Rand presented Objectivism as a philosophy of reason. But is it? That is the question Scott Ryan seeks to answer in this careful examination of the Objectivist epistemology and its alleged sufficiency as the philosophical foundation of a free and prosperous commonwealth. Sorting painstakingly through Rand’s writings on the subject, Mr. Ryan concludes that the epistemology of Objectivism is incoherent and debases both the concept and the practice of rationality.

[Note: Some of the formatting of the text has been corrupted and differs from the print version]

Scott Ryan has been a guitarist, singer, and songwriter, a mathematics teacher, a writer and editor, a computer programmer, a Sherlockian, and a husband and father. He holds a master’s degree in mathematics and a juris doctor, and has had a lifelong interest in philosophy. He currently works as a software developer and lives in Cuyahoga Falls, Ohio, with his wife.

Much ado about Nothing: Robert Spitzer vs. Stephen Hawking on the creation of the Universe

Edit: Please excuse the weird formatting, I've tried as hard as I can to fix it and nothing seems to work.

(A caveat: I have not read either of Hawking’s or Spitzer’s recent books, so my critique of Spitzer’s argument is based only on his blog post.)

In his blog post “The curious metaphysics of Dr. Stephen Hawking”, Fr. Robert Spitzer offers some criticisms of Stephen Hawking’s recent book The Grand Design. I agree with some of his criticisms, but not with others. First, a point of agreement. Fr. Spitzer addresses the following quote from Hawking’s book:

“Because there is a law such as gravity, the Universe can and will create itself from nothing. Spontaneous creation is the reason there is something rather than nothing, why the Universe exists, why we exist.”

Fr. Spitzer points out—rightly, in my view—that if the law of gravity is to account for the creation of the universe, it must exist, and hence the creation of the universe is not really “from nothing”. Furthermore, I agree with him when he says,

“[…] these thinkers [i.e., Parmenides and Plato] use the term “nothing” to mean “nothing” (i.e. “that which there is no such thing as”). Nothing should not be thought to be a vacuum or a void (which is dimensional and orientable – where you can have more or less space); and it is certainly not a physical law.”

If we understand the term ‘universe’ broadly, as including physical laws, then indeed we cannot explain why the universe exists by reference to such laws, nor can we use physical laws to explain why there is something rather than nothing, for they must exist in order to explain, and so we would only have explained the existence of “something” by the existence of “something else”.

However, Fr. Spitzer does not merely criticize Hawking’s claim. He also makes the following argument for the thesis that “only nothing can come from nothing”, which he hopes will show that the physical universe must have a transcendent cause:

“But let’s go back to Dr. Hawking’s underlying assumption, namely that there are reasons to think that something came from nothing – namely, reasons for a beginning. How have philosophers and metaphysicians traditionally responded to this question? With what many term the first principle of metaphysics, “From nothing only nothing comes.” If you take nothing literally – that is if one acknowledges that there is no such thing as nothing, then one cannot attribute anything to nothing. One cannot attribute characteristics, actions, powers and so forth to nothing. In this absence of everything, one can only conclude that “only nothing can come from nothing.” What does this mean?

It means that if the physical universe had a beginning (a point at which it came into existence” then prior to that point it was nothing. And if it was nothing then it could not have created itself (because only nothing can come from nothing). So what does that imply? The very reality that Dr. Hawking wants to avoid, namely, a transcendent power which can cause the universe to come into existence.”

Now we have come to the point where I must disagree. If one is committed to the thesis that there is no such thing as nothing, it is of course true that one cannot attribute anything to “it.” But that is not to say that there is some mysterious metaphysical somewhat which we have decided to call ‘nothing’, and which somehow “is” without being a “thing” or without having any attributes. Instead, to say that there is no such thing as nothing is just to say that there is no object which is not the same thing as some object, and/or that there is no object which does not have attributes, where the word ‘object’ applies not only to physical objects, but to everything that exists.

While I doubt that Fr. Spitzer holds the view that “nothing” is a mysterious metaphysical somewhat, the phrasing of the passage I quoted above does suggest it. For he says, “if the physical universe had a beginning (a point at which it came into existence” then prior to that point it was nothing. And if it was nothing then it could not have created itself (because only nothing can come from nothing).” That suggests that there is something that the universe was prior to its beginning—namely “nothing”! But if one rejects the “metaphysical somewhat” conception of “nothing”, one should say, more perspicuously, that prior to the beginning of the universe it was not the case that anything existed. Furthermore, if it was not the case that anything existed, it also was not the case that anything existed which could have created the universe. That being so, it follows straightaway that the universe could not have created itself, there being no universe around to do the creating. If this is what Fr. Spitzer means, then I think he is surely right. But then he has not really shown that “only nothing can come from nothing”, only that the universe could not have created itself from nothing. There remains another possibility which he has not yet foreclosed, which is that the universe had a beginning without being created by anything. We may say that nothing caused the universe to come into existence, not in the sense that its coming-into-existence was caused by a mysterious metaphysical somewhat called ‘nothing’, but rather in the sense that it came into existence without having any cause at all. Fr. Spitzer may have other arguments to show why the universe must have a transcendent cause, and for all I know he may be right that it has one. I only take myself to have shown that the thesis that “only nothing can come from nothing” isn’t going to do the trick.

Lewis on Devitt on Ostrich Nominalism

In his “New Work for a Theory of Universals”[1], David Lewis discusses Michael Devitt’s defense of Ostrich Nominalism in his article “‘Ostrich Nominalism’ or ‘Mirage Realism’?”[2], specifically as a response to the One Over Many argument. Devitt had proposed to paraphrase such sentences as

“a and b have the same property, F-ness”

as

“a and b are both F”

which itself can be analyzed as

“a is F”

and

“b is F”.

Lewis thinks that this is not satisfactory. He says:

But Devitt has set himself too easy a problem. If we attend to the modest, untransformed One over Many problem, which is no mirage, we will ask about a different analysandum:

a and b have some common property (are somehow of the same type)

in which it is not said what a and b have in common. This less definite analysandum is not covered by what Devitt has said.[3]

I think there is an obvious paraphrase of Lewis’ example which, though not explicitly covered by what Devitt had said, is in perfect harmony with its spirit. Indeed, I think it’s obvious enough that it's probable someone else has already thought of it, which for a while made me hesitant to make this post. Nevertheless, I’m interested to see if others think the paraphrase works, so I’m posting this anyway, even if I can’t claim originality for it. The paraphrase goes like this (where F, G, H, etc., are all the predicates expressible in the language):

((Fa & Fb) v [(Ga &Gb) v (Ha &Hb)])…

Or, in English:

Either a and b are both F or a and b are both G or a and b are both H…

So my question is: Do you think the paraphrase works? And if not, why not?

---

[1] As reprinted in Properties (Oxford Readings in Philosophy), edited by D. H. Mellor and Alex Oliver. Oxford University Press, 1997.

[2] Reprinted in the same volume as in the above footnote.

[3] See p. 201 of Properties (Oxford Readings in Philosophy).

Points and Platonism

In this post I want to discuss the ontological status of points and its significance for semantic arguments for Platonism. No, not geometrical points; the points I’m talking about are the ones you gain or lose in playing certain kinds of games (including sports).

Suppose a Platonist argues as follows:

Let’s say that John has ten points in a game. Surely, then, these points must exist: Nothing is true of the non-existent, so if John really has ten points, there must be ten points which John has. Or consider the statement: “John is two points ahead of Fred”. This seems to assert a relation between the number of points John has and the number of points Fred has. If one thing stands in a relation to another they both must exist, so if “John is two points ahead of Fred” is true there must be points which each of them has. “And if one holds that points don’t exist,” the Platonist might add, “who should be the one to tell poor John and Fred that they both lose because they have no points?”

What is one to think of such reasoning? I have chosen this example because it resembles the sort of semantic arguments Platonists often use in other domains, in particular semantic arguments for properties and mathematical objects. In this case, however, the nature of the objects posited is particularly problematic. First, it seems difficult to conceive what these points would be: Besides trivially essential properties—properties such as being identical to itself, which are shared by everything that exists—and those such as being a point belonging to John, what properties would points have? This might not be so bad in itself, for there are more respectable abstract objects--sets, for example--which also seem to have no properties besides the kind mentioned above and those attributed to them by set theory. But one can ask more troubling questions. For example, if John loses a point and Fred gains one, is the point John lost the same point as the one Fred gained, or a different one? If questions of identity and difference make no sense for points, I think that is a good (though arguably not indefeasible) reason for rejecting such entities. But the plight of points is worse yet: Suppose John and Fred are playing a game where, so the rules prescribe, it is possible to have a negative number of points. We can imagine that they are playing a board game where you roll dice and move your piece the indicated number of spaces, and in which landing on a penalty box costs a player five points even when the number of points they have is less than five. The player who has the most points after twenty moves have been made wins. So, for example, if John has -25 points and Fred has -10 after twenty moves, Fred wins the game. Should the Platonist conclude from this example that, since points really exist, it is possible to have a negative number of something? Or should they say instead that, since there can’t be a negative number of anything, this game is somehow illegitimate?

As with other abstract objects, there are also epistemological problems with accepting the existence of points. If points are independently existing entities, how can we be sure we really gain or lose them in the manner the rules of the game prescribe? Could it be that, if the rules of the game say that move m is worth five points, one might only gain three points on executing move m because of some ontological glitch? For Humeans who reject necessary connections between distinct existences, it should appear suspicious that anything could guarantee that something we do causes us to stand in some contingent (or “external”) relation to independently existing abstracta.

If one finds the prospect of answering these bizarre questions distasteful, one will want to find some way of rejecting points while retaining the ability to make sense of our game-related talk and behavior. One might seek a nominalistic paraphrase of statements ostensibly about points, or perhaps try to give a Wittgensteinian account in which our talk of points is treated as moves in a language-game which is practically indispensable to our keeping score. Whatever one says here, I think it is clear that the semantic argument for points faces serious difficulties. If it can’t be made to work here, why should it fare any better with respect to other kinds of abstract objects? I wouldn’t say that other sorts of abstract objects can’t exist—on the contrary, I think some do. But if they do, it seems to me, we need far better arguments for believing in them.